Rectified approximations for the solution of nonlinear equations

摘要

In this paper we introduce the ‘rectification method’ for the construction of algorithms of pre-designed order r for the solution of nonlinear equations f(x) = 0. Our method is based upon the derivation of a rectified approximation g(x) to f(x), via Padé formulas, such that the application of the Newton-Raphson iterations to g generates the desired rth order algorithm. Various properties of g are explored as are recursive relations among rectified approximations associated with successive orders of convergence. It is demonstrated that the use of g in favor of f can relax standard sufficient conditions assuring convergence of the iterations.